On some inequalities for weighted products and ratios of the sine function

Résumé

We introduce a reflection-substitution technique for sine inequalities that yields, via Hölder's inequality and its reverse, $2^{n-1}$ distinct upper bounds for products and lower bounds for ratios of weighted sines. The comparison between bounds reduces to simple sign conditions on linear combinations of the variables. We fully classify the two- and three-dimensional cases, providing explicit dominance criteria with detailed examples for each signature.

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